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math.NT — Number Theory

An explicit upper-bound certificate for the essential minimum of the Zhang–Zagier height

Contributed by Ricardo Pinho

We give an exact certificate proving $C_{82}<0.2534449$ for the essential minimum of $h(\alpha)+h(1-\alpha)$ , where $h$ is the absolute logarithmic Weil height. We use Doche's limit-point method, building on Grinsztajn's factor library and, in particular, Épi's unequal-multiplicity construction. We retain Épi's denominator factors, add Doche's $P_7$ , and add 25 numerator factors found by LP-dual-weighted evaluation lattices with LLL. Joint linear-program re-optimisation is followed by exact rational rounding. Two separately written ball-arithmetic checkers certify the bound; a further adversarial computational audit accepts the certificate.

Primarily AI-generated textHuman understanding: some partsDoche limit-point methodLLLWeil heightZhang-Zagier heightcomputer-assisted proofessential minimumlinear programming

Oct 21, 2026PDF ↗View work →

hexagon:2610.00425v1

math.NT — Number Theory

An explicit upper-bound certificate for the Schur–Siegel–Smyth trace constant

Contributed by Ricardo Pinho

We document an explicit certificate proving $C_{86}<1.8117251$ for the smallest limiting trace-to-degree ratio of totally positive algebraic integers. The construction uses the primal problem of Orloski and Talebizadeh Sardari, with their four-polynomial set enlarged to eleven. A discretised solution is rounded to a rational step density, then normalised to a probability measure. Exact rational calculations and rigorous ball-arithmetic inequalities verify Smith's sufficient criterion for an arithmetic measure. An independently written agent checker accepts the certificate and rejects four corrupted controls. This note records the proof, data provenance, reproduction results and limitations of the previously released certificate; it makes no optimality claim.

Primarily AI-generated textHuman understanding: some partsSchur-Siegel-Smyth trace problemball arithmeticcomputer-assisted prooflinear programmingtotally positive algebraic integers

Oct 21, 2026PDF ↗View work →

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An explicit upper-bound certificate for the essential minimum of the Zhang–Zagier height

An explicit upper-bound certificate for the Schur–Siegel–Smyth trace constant

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